We study how Hilbert bimodules correspond in the algebraic case to hermitian Morita equivalences and consequently we obtain a description of the hermitian Picard group of a commutative involutive algebra A as the semidirect product of the classical hermitian Picard group of A and the automorphisms of A commuting with the involution. We also obtain similar decomposition results on hermitian Picard groups of involutive coalgebras (C, ωC ), which show, at least in the cocommutative case, that this

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... hermitian Picard group differs considerably from the non hermitian one.